Finite Group Databasebeta

← #6471064

Subgroups — #6471064

Order 512.

1824 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35014C211yesmaximal, characteristic
2C2×#2360C211yesmaximal, characteristic
3#33149C211yesmaximal, characteristic, complemented
4#33109C211yesmaximal, characteristic, complemented
5#32983C212yesmaximal
6#29953C211yesmaximal, characteristic, complemented
7#29715C211yesmaximal, characteristic, complemented
8#29597C212yesmaximal
9#27036C214yesmaximal, complemented
10#26726C214yesmaximal, complemented
11#26621C214yesmaximal, complemented
12#26559C214yesmaximal, complemented
13#22079C212yesmaximal, complemented
14#21902C211yesmaximal, characteristic
15C4×#451C212yesmaximal, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C2×#484C2212yes—
20C2×#484C2214yes—
21C2×#451C2212yes—
22C2×C4×Q16C2214yescomplemented
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25#2848C2212yes—
26#2646—21no—
27#2626—21no—
28#2589—22no—
29#2587—22no—
30#2497C2212yes—
31#2492C2212yes—
32#2429C2212yes—
33#2423C2212yes—
34#2360—22no—
35#2360—22no—
36#2360C2214yes—
37#2360C2214yes—
38#2329—21no—
39#2312—21no—
40C4×C22⋊Q8C2212yes—
41C2×#391C2211yescharacteristic
42C2×#389—21no—
43C2×#389C2211yescharacteristic
44C2×#387—21no—
45C2×#385—21no—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4—21no—
48C2×C4×C22⋊C4C2211yescharacteristic
49#2095C2212yescomplemented
50#2072C2212yescomplemented
51#2068C2214yescomplemented
52#2056C2214yescomplemented
53#2036C2214yes—
54#2036C2214yes—
55#2024C2212yescomplemented
56#2016C2212yescomplemented
57#2001C2214yes—
58#2001C2214yes—
59#1987C2212yescomplemented
60#1981C2212yescomplemented
61#1947C2212yescomplemented
62#1947C2212yescomplemented
63#1942C2214yescomplemented
64#1933C2214yescomplemented
65#1928C2214yescomplemented
66#1923C2214yes—
67#1923C2214yes—
68#1921C2214yescomplemented
69#1921C2214yescomplemented
70#1878C2214yescomplemented
71#1872C2212yes—
72#1872C2212yes—
73#1872C2214yes—
74#1863C2214yescomplemented
75#1863C2214yescomplemented
76#1852C2212yes—
77#1852C2212yes—
78#1852C2214yes—
79#1824C2212yes—
80#1824C2214yes—
81#1797C2211yescharacteristic
82#1797C2211yescharacteristic
83#1787C2212yes—
84#1779C2212yes—
85C4×#103C2214yescomplemented
86C4×C22⋊C8C2212yes—
87C23×Q8C2311yescharacteristic
88C24×C4—21no—
89C24×C4C2311yescharacteristic
90C22×Q16C2314yes—
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8C2312yes—
98C2×C22⋊Q8C2314yes—
99C2×C22⋊Q8C2312yes—
100C2×C22⋊Q8C2312yes—
101C2×C4×Q8—22no—
102C2×C4×Q8—44no—
103C2×C4×Q8C2314yes—
104C2×C4×Q8C2312yes—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—22no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—44no—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4—21no—
116C22×C22⋊C4D411yescharacteristic
117C22×C22⋊C4D411yescharacteristic
118C22×C42—21no—
119C22×C42—21no—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#484—28no—
123#484—48no—
124#451—28no—
125C4×Q16—48no—
126C2×#107C2314yes—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2312yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2314yes—
140C2×C22⋊C8C2314yes—
141C2×C4×C8C2314yes—
142#400—22no—
143#400—22no—
144#399—22no—
145#398—22no—
146#398—22no—
147#396—22no—
148#396—22no—
149#393—22no—
150#391—21no—
151#391—21no—
152#391—21no—
153#391—22no—
154#391C2312yes—
155#391C2312yes—
156#391C2312yes—
157#391C2312yes—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—22no—
162#389—22no—
163#389—24no—
164#389—24no—
165#389C2312yes—
166#389C2314yes—
167#389C2312yes—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—24no—
176#387—24no—
177#385—22no—
178#385—22no—
179#385—22no—
180#385—22no—
181#385—24no—
182#385—24no—
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#380—21no—
186#380—21no—
187#380—21no—
188#380C2312yes—
189#380C2312yes—
190#380C2312yes—
191#379—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4C2312yes—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—24no—
200C4×C22⋊C4—24no—
201C4×C22⋊C4D412yes—
202C4×C22⋊C4D412yes—
203C4×C22⋊C4C2312yes—
204C4×C22⋊C4C2312yes—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95D411yescharacteristic
210C2×#95D411yescharacteristic
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2312yes—
218#336C2312yes—
219#336C2314yes—
220C25C2×D411yescharacteristic
221C22×Q8—21no—
222C22×Q8—21no—
223C22×Q8—22no—
224C22×Q8—42no—
225C22×Q8—42no—
226C22×Q8#4114yes—
227C22×Q8C2412yes—
228C22×Q8C2412yes—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—21no—
235C23×C4—24no—
236C23×C4—42no—
237C23×C4—42no—
238C23×C4—44no—
239C23×C4—44no—
240C23×C4—82no—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D412yes—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2412yes—
248C23×C4C2411yescharacteristic
249C2×Q16—28no—
250C2×Q16—44no—
251C2×Q16—44no—
252C22×C8C2414yes—
253C22×C8C2414yes—
254C4⋊Q8—42no—
255C4⋊Q8—42no—
256C22⋊Q8—216no—
257C22⋊Q8—28no—
258C22⋊Q8—28no—
259C22⋊Q8—24no—
260C22⋊Q8—24no—
261C22⋊Q8—44no—
262C22⋊Q8—44no—
263C22⋊Q8—44no—
264C4×Q8—28no—
265C4×Q8—44no—
266C4×Q8—44no—
267C4×Q8—48no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—48no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—82no—
298C2×C4⋊C4#4114yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4Q1618yescomplemented
315C2×C22⋊C4C2×D412yes—
316C2×C22⋊C4C2×D412yes—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—42no—
325C2×C42—42no—
326C2×C42—44no—
327C2×C42—82no—
328C2×C42C2×D412yes—
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2412yes—
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334#107—28no—
335#107—28no—
336#107—44no—
337C4⋊C8—42no—
338C4⋊C8—42no—
339#103—28no—
340#103—28no—
341#103—28no—
342#103—44no—
343#103—44no—
344#103—44no—
345C22⋊C8—24no—
346C22⋊C8—24no—
347C22⋊C8—44no—
348C22⋊C8—44no—
349C4×C8—44no—
350#95—24no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95C2×D412yes—
368#95C2×D412yes—
369#95C2×D412yes—
370#95C2×D412yes—
371C24—22no—
372C24—22no—
373C24—41no—
374C24—41no—
375C24—41no—
376C24—41no—
377C24C2×Q1614yes—
378C24C22×D412yes—
379C24C22×D411yescharacteristic
380C2×Q8—24no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—24no—
385C2×Q8—24no—
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—48no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—48no—
392C2×Q8—42no—
393C2×Q8—81no—
394C2×Q8—81no—
395C2×Q8C2×#4114yes—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—42no—
421C22×C4—41no—
422C22×C4—42no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—48no—
441C22×C4—42no—
442C22×C4—84no—
443C22×C4—816no—
444C22×C4C2×Q1614yes—
445C22×C4C2×Q1614yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2511yescharacteristic
457Q16—416no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C4⋊C4—24no—
465C4⋊C4—216no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—81no—
481C4⋊C4—81no—
482C4⋊C4—82no—
483C4⋊C4—81no—
484C4⋊C4—84no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—81no—
489C4⋊C4—81no—
490C4⋊C4—81no—
491C4⋊C4C2×#4114yes—
492C4⋊C4H(5,𝔽2)14yes—
493C22⋊C4—216no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C22⋊C4C2×Q1618yes—
503C22⋊C4C2×Q1618yes—
504C42—24no—
505C42—22no—
506C42—22no—
507C42—41no—
508C42—41no—
509C42—42no—
510C42—48no—
511C42—82no—
512C42—82no—
513C42—82no—
514C42—82no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—22no—
520C23—22no—
521C23—28no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23#52512yes—
533C23C22×Q1614yes—
534C23C22×Q1612yes—
535C23#57512yes—
536C23C23×D411yescharacteristic
537Q8—28no—
538Q8—48no—
539Q8—44no—
540Q8—44no—
541Q8—44no—
542Q8—162no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—44no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—816no—
601C2×C4—84no—
602C2×C4#52512yes—
603C2×C4C22×Q1612yes—
604C2×C4C22×Q1612yes—
605C2×C4C22×Q1614yes—
606C2×C4C22×#4111yescharacteristic
607C2×C4C22×#4111yescharacteristic
608C2×C4C2×H(5,𝔽2)12yes—
609C2×C4#58512yes—
610C8—44no—
611C8—44no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—416no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22#310414yes—
627C22C2×#52511yescharacteristic
628C22#348411yescharacteristic
629C22#348511yescharacteristic
630C22C23×Q1611yescharacteristic
631C22C2×#57511yescharacteristic
632C22#358511yescharacteristic
633C22#358611yescharacteristic
634C4—22no—
635C4—22no—
636C4—24no—
637C4—22no—
638C4—41no—
639C4—41no—
640C4—42no—
641C4—44no—
642C4—44no—
643C4—42no—
644C4—44no—
645C4—82no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4—82no—
650C4#345912yes—
651C4#350612yes—
652C2—24no—
653C2—44no—
654C2#3498611yescharacteristic
655C2C2×#310411yescharacteristic
656C2#6056511yescharacteristic
657C2#6156311yescharacteristic
658C2#6157611yescharacteristic
659C2#6169811yescharacteristic
660C2#6171611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #33149C2semidirect
4. #33109C2semidirect
6. #29953C2semidirect
7. #29715C2semidirect
9. #27036C2semidirect
10. #26726C2semidirect
11. #26621C2semidirect
12. #26559C2semidirect
13. #22079C2semidirect
15. C4×#451C2semidirect
22. C2×C4×Q16C22semidirect
49. #2095C22semidirect
50. #2072C22semidirect
51. #2068C22semidirect
52. #2056C22semidirect
55. #2024C22semidirect
56. #2016C22semidirect
59. #1987C22semidirect
60. #1981C22semidirect
61. #1947C22semidirect
62. #1947C22semidirect
63. #1942C22semidirect
64. #1933C22semidirect
65. #1928C22semidirect
68. #1921C22semidirect
69. #1921C22semidirect
70. #1878C22semidirect
74. #1863C22semidirect
75. #1863C22semidirect
85. C4×#103C22semidirect
314. C2×C22⋊C4Q16semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#350142
C2×#23602
#331492
#331092
#329832
#299532
#297152
#295972
#270362
#267262
#266212
#265592
#220792
#219022
C4×#4512